{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Geometric multigrid solvers\n",
    "\n",
    "In addition to the full gamut of algebraic solvers offered by PETSc, Firedrake also provides access to multilevel solvers with geometric hierarchies. In this tutorial, we will study strategies to solve the Stokes equations, demonstrating how the multigrid functionality composes with fieldsplit preconditioning.\n",
    "\n",
    "\n",
    "## Creating a geometric hierarchy\n",
    "\n",
    "Geometric multigrid requires a geometric hierarchy of meshes on which the equations will be discretised.  For now, Firedrake supports hierarchies of *regularly refined* meshes, which we create by providing a *coarse mesh* and building a `MeshHierarchy`.  This hierarchy encapsulates the relationship between coarse and fine cells."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib notebook\n",
    "import matplotlib.pyplot as plt\n",
    "from firedrake import *\n",
    "\n",
    "coarse_mesh = RectangleMesh(15, 10, 1.5, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having made the coarse mesh, we create the hierarchy of meshes.  The second argument tells Firedrake how many levels of refinement to use.  Here we refine three times, so that in total we have four meshes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "hierarchy = MeshHierarchy(coarse_mesh, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `hierarchy` object behaves like a Python *iterable*, so we can ask for its length and index it to extract meshes on a given level in the normal way:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(hierarchy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "finest_mesh = hierarchy[-1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Grid transfer\n",
    "\n",
    "If you want to control all aspects of the multigrid cycle, Firedrake offers the necessary building blocks.  You just need to create the relevant objects on the levels of the mesh hierarchy, and then use provided functions to transfer information between levels.\n",
    "\n",
    "Firedrake provides the three functions, `prolong`, `restrict`, and `inject`.\n",
    "\n",
    "- `prolong` transfers primal quantities from coarse to fine meshes.\n",
    "- `restrict` transfers dual quantities (residuals) from fine to coarse meshes.  It is the dual of `prolong`.\n",
    "- `inject` transfers primal quantities from fine to coarse meshes.\n",
    "\n",
    "Most of the time, there is no need to access the interface at this level.  Instead, it suffices to define the variational problem on the finest level of a mesh hierarchy and then drive the solver using PETSc options.  This is the most flexible method, which we now demonstrate."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise\n",
    "\n",
    "Create mesh hierarchy of two levels on an interval mesh.  Create a piecewise linear function on the fine mesh and interpolate the function $f(x) = x$.  Now use the grid transfer functionality to move this function to the coarse mesh (using both `restrict` and `inject`).  Do you notice a difference between the two outcomes?\n",
    "\n",
    "- Hint 1: You will need to create a `FunctionSpace` on both the coarse and fine mesh.\n",
    "- Hint 2: use `x, = SpatialCoordinate(mesh)` to gain access to the `x` coordinate on a given mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problem setup\n",
    "\n",
    "We will solve the Stokes equations on a rectangular domain $\\Omega = [0, 1.5] \\times [0, 1]$.  With constant inflow and outflow through two \"pipes\" and no slip boundaries everywhere else.\n",
    "\n",
    "Our problem is to find $(u, p) \\in V\\times Q$ such that:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "-\\nu \\nabla^2 u + \\nabla p &= 0 \\quad \\text{in $\\Omega$}\\\\\n",
    "\\nabla \\cdot u &= 0 \\quad \\text{in $\\Omega$}\\\\\n",
    "u &= u_0 \\quad \\text{on $\\Gamma_{\\text{inout}}$}\\\\\n",
    "u &= (0, 0) \\quad \\text{on $\\Gamma \\setminus \\Gamma_{\\text{inout}}$}\n",
    "\\end{align}\n",
    "$$\n",
    "where \n",
    "$$\n",
    "\\Gamma_\\text{inout}(x, y) = \\{(x, y)\\, |\\, y \\in [1/6, 1/3] \\cup [2/3, 5/6], x \\in \\{0, 1.5\\} \\}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "u_0(x, y) = \\bigg\\{\\\n",
    "\\begin{split} \n",
    "1 - (12 (y - 1/4))^2 \\quad y &< 1/2 \\\\\n",
    "1 - (12 (y - 3/4))^2 \\quad y &> 1/2 \\\\\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "We will use Taylor-Hood elements.  In the usual way, we multiply by test functions and after integrating by parts and incorporating boundary conditions, we arrive at the weak formulation, find $(u, p) \\in V \\times Q$ such that\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\int_\\Omega \\nu \\nabla u : \\nabla v - p\\nabla \\cdot v\\,\\text{d}x &= 0 \\quad \\forall v \\in V\\\\\n",
    "\\int_\\Omega q \\nabla \\cdot u\\,\\text{d}x &= 0 \\quad \\forall q \\in Q \\\\\n",
    "u &= (1, 0) \\quad \\text{on $\\Gamma_{\\text{inout}}$}\\\\\n",
    "u &= (0, 0) \\quad \\text{on $\\Gamma \\setminus \\Gamma_{\\text{inout}}$}.\n",
    "\\end{align}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implementation\n",
    "\n",
    "To make things easier to play with, we'll wrap everything up in a function that we can call to produce a solver."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_solver(parameters=None):\n",
    "    coarse_mesh = RectangleMesh(15, 10, 1.5, 1)\n",
    "    hierarchy = MeshHierarchy(coarse_mesh, 3)\n",
    "    \n",
    "    mesh = hierarchy[-1]\n",
    "    \n",
    "    V = VectorFunctionSpace(mesh, \"CG\", 2)\n",
    "    Q = FunctionSpace(mesh, \"CG\", 1)\n",
    "    W = V*Q\n",
    "    \n",
    "    u, p = TrialFunctions(W)\n",
    "    v, q = TestFunctions(W)\n",
    "    \n",
    "    nu = Constant(1)\n",
    "    x, y = SpatialCoordinate(mesh)\n",
    "    \n",
    "    t = conditional(y < 0.5, y - 1/4, y - 3/4)\n",
    "    gbar = conditional(Or(And(1/6 < y,\n",
    "                              y < 1/3),\n",
    "                          And(2/3 < y,\n",
    "                              y < 5/6)),\n",
    "                       1, \n",
    "                       0)\n",
    "\n",
    "    value = as_vector([gbar*(1 - (12*t)**2), 0])\n",
    "    bcs = [DirichletBC(W.sub(0), interpolate(value, V), (1, 2)),\n",
    "           DirichletBC(W.sub(0), zero(2), (3, 4))]\n",
    "    \n",
    "    a = (nu*inner(grad(u), grad(v)) - p*div(v) + q*div(u))*dx\n",
    "    L = inner(Constant((0, 0)), v)*dx\n",
    "    wh = Function(W)\n",
    "    problem = LinearVariationalProblem(a, L, wh, bcs=bcs)\n",
    "    solver = LinearVariationalSolver(problem, solver_parameters=parameters, appctx={\"nu\": nu})\n",
    "    return solver"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "solver = create_solver({\"ksp_type\": \"preonly\",\n",
    "                        \"pc_type\": \"lu\",\n",
    "                        \"pc_factor_shift_type\": \"inblocks\",\n",
    "                        \"ksp_monitor\": None,\n",
    "                        \"pmat_type\": \"aij\"})\n",
    "solver.solve()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can extract the solution variable from the solver object and plot the velocity and pressure fields.\n",
    "In previous notebooks, we've seen how to create multiple subplots, arrange them within a figure, share the axes, set the aspect ratio, and add titles.\n",
    "Here we're also taking the output of each plotting function -- a set of streamlines from streamplot and a set of colored triangles from tripcolor -- and using them to add a colorbar to each subplot.\n",
    "When we create the colorbar, we also need to tell matplotlib which axis to draw it on.\n",
    "Finally, we're adjusting the size and spacing of the colorbar to make the result look nicer.\n",
    "Steps like this often require some trial and error, but they're essential for making publication-quality figures."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
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       "              'Firefox 4 and 5 are also supported but you ' +\n",
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       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
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       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
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       "    var canvas_div = $('<div/>');\n",
       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
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       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
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       "        stop: function(event, ui) {\n",
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       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
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       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
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       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
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       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img src=\"\" width=\"852.75\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# NBVAL_IGNORE_OUTPUT\n",
    "w = solver._problem.u\n",
    "u, p = w.split()\n",
    "fig, axes = plt.subplots(ncols=2, sharex=True, sharey=True)\n",
    "streamlines = streamplot(u, resolution=1/30, seed=4, axes=axes[0])\n",
    "axes[0].set_aspect(\"equal\")\n",
    "axes[0].set_title(\"Velocity\")\n",
    "fig.colorbar(streamlines, ax=axes[0], fraction=0.032, pad=0.02)\n",
    "\n",
    "triangles = tripcolor(p, axes=axes[1], cmap='coolwarm')\n",
    "axes[1].set_aspect(\"equal\")\n",
    "axes[1].set_title(\"Pressure\")\n",
    "fig.colorbar(triangles, ax=axes[1], fraction=0.032, pad=0.02);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This direct method is not a scalable solution technique for large problems.  Similar to our earlier example involving the mixed Poisson problem, a Schur complement method can be more efficient.  We'll use geometric multigrid to invert the elliptic velocity block, and use a viscosity-weighted pressure mass matrix to precondition the Schur complement.  This gives good results as long as viscosity contrasts are not too strong. The Python preconditioner that we use to create the mass matrix is described in more detail in the composable solvers notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "class MassMatrix(AuxiliaryOperatorPC):\n",
    "    def form(self, pc, test, trial):\n",
    "        # Grab the viscosity\n",
    "        nu = self.get_appctx(pc)[\"nu\"]\n",
    "        return (nu*test*trial*dx, None)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "parameters = {\n",
    "    \"ksp_type\": \"gmres\",\n",
    "    \"ksp_monitor\": None,\n",
    "    \"pc_type\": \"fieldsplit\",\n",
    "    \"pc_use_amat\": True,\n",
    "    \"pc_fieldsplit_type\": \"schur\",\n",
    "    \"pc_fieldsplit_schur_fact_type\": \"lower\",\n",
    "    \"fieldsplit_0_ksp_type\": \"preonly\",\n",
    "    \"fieldsplit_0_pc_type\": \"mg\",\n",
    "    \"fieldsplit_1_ksp_type\": \"preonly\",\n",
    "    \"fieldsplit_1_pc_type\": \"python\",\n",
    "    \"fieldsplit_1_pc_python_type\": \"__main__.MassMatrix\",\n",
    "    \"fieldsplit_1_aux_pc_type\": \"icc\",\n",
    "}\n",
    "\n",
    "solver = create_solver(parameters)\n",
    "solver.solve()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another option is to use a distributive smoother.  Instead of using a Schur complement on the outside and multigrid for the velocity block, we can instead use multigrid on the outside and Schur complements as a \"smoother\" on each level.  This requires more parameters, but no other change in our problem setup.\n",
    "\n",
    "Notice how we provide the `MassMatrix` Python preconditioner on each level for the Schur complement. An appropriate operator will be created on each level as necessary."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "firedrake:WARNING Creating new TransferManager to transfer data to coarse grids\n",
      "firedrake:WARNING This might be slow (you probably want to save it on an appctx)\n"
     ]
    }
   ],
   "source": [
    "parameters = {\n",
    "      \"ksp_type\": \"fgmres\",\n",
    "      \"ksp_monitor\": None,\n",
    "      \"mat_type\": \"nest\",\n",
    "      \"pc_type\": \"mg\",\n",
    "      \"mg_coarse_ksp_type\": \"preonly\",\n",
    "      \"mg_coarse_pc_type\": \"fieldsplit\",\n",
    "      \"mg_coarse_pc_fieldsplit_type\": \"schur\",\n",
    "      \"mg_coarse_pc_fieldsplit_schur_fact_type\": \"full\",\n",
    "      \"mg_coarse_fieldsplit_0_ksp_type\": \"preonly\",\n",
    "      \"mg_coarse_fieldsplit_0_pc_type\": \"lu\",\n",
    "      \"mg_coarse_fieldsplit_1_ksp_type\": \"richardson\",\n",
    "      \"mg_coarse_fieldsplit_1_ksp_richardson_self_scale\": True,\n",
    "      \"mg_coarse_fieldsplit_1_ksp_max_it\": 5,\n",
    "      \"mg_coarse_fieldsplit_1_pc_type\": \"none\",\n",
    "      \"mg_levels_ksp_type\": \"richardson\",\n",
    "      \"mg_levels_ksp_richardson_self_scale\": True,\n",
    "      \"mg_levels_ksp_max_it\": 1,\n",
    "      \"mg_levels_pc_type\": \"fieldsplit\",\n",
    "      \"mg_levels_pc_fieldsplit_type\": \"schur\",\n",
    "      \"mg_levels_pc_fieldsplit_schur_fact_type\": \"upper\",\n",
    "      \"mg_levels_fieldsplit_0_ksp_type\": \"preonly\",\n",
    "      \"mg_levels_fieldsplit_0_pc_type\": \"bjacobi\",\n",
    "      \"mg_levels_fieldsplit_0_sub_pc_type\": \"ilu\",\n",
    "      \"mg_levels_fieldsplit_1_ksp_type\": \"preonly\",\n",
    "      \"mg_levels_fieldsplit_1_pc_type\": \"python\",\n",
    "      \"mg_levels_fieldsplit_1_pc_python_type\": \"__main__.MassMatrix\",\n",
    "      \"mg_levels_fieldsplit_1_aux_pc_type\": \"icc\",\n",
    "}\n",
    "\n",
    "solver = create_solver(parameters)\n",
    "solver.solve()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise\n",
    "\n",
    "In most real-world scenarios, the viscosity $\\nu$ will not be constant, but rather variable.  See what happens to the performance of the solver when you replace the constant viscosity $\\nu$ with a spatially varying one.  In particular, try:\n",
    "\n",
    "$$\n",
    "\\nu(x, y) = \\bigg\\{\\begin{split}\n",
    "&100 \\quad&\\text{if $(x - 0.5)^2 + (y - 0.5)^2 < 0.25$}\\\\\n",
    "&1 \\quad&\\text{otherwise.}\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "- Hint 1: Use `conditional` to produce an expression that varies the viscosity spatially.\n",
    "- Hint 2: You can determine the iteration count after the solver has finished using `solver.snes.ksp.getIterationNumber()`.\n",
    "\n",
    "Here are some questions you might consider:\n",
    "\n",
    "- What happens to the iteration count when you increase the number levels in the hierarchy?\n",
    "\n",
    "- What if you change the viscosity contrast to 1000, or 10000?\n",
    "\n",
    "- Compare with algebraic multigrid to solve the velocity block (use `\"fieldsplit_0_pc_type\": \"hypre\"` instead of `\"fieldsplit_0_pc_type\": \"mg\"`).\n",
    "\n",
    "For simplicity, the relevant setup is copied below to start with:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def create_solver(parameters=None):\n",
    "    coarse_mesh = RectangleMesh(15, 10, 1.5, 1)\n",
    "    hierarchy = MeshHierarchy(coarse_mesh, 3)\n",
    "    \n",
    "    mesh = hierarchy[-1]\n",
    "    \n",
    "    V = VectorFunctionSpace(mesh, \"CG\", 2)\n",
    "    Q = FunctionSpace(mesh, \"CG\", 1)\n",
    "    W = V*Q\n",
    "    \n",
    "    u, p = TrialFunctions(W)\n",
    "    v, q = TestFunctions(W)\n",
    "    \n",
    "    # Change me to spatially varying.\n",
    "    nu = Constant(1)\n",
    "    x, y = SpatialCoordinate(mesh)\n",
    "    \n",
    "    t = conditional(y < 0.5, y - 1/4, y - 3/4)\n",
    "    gbar = conditional(Or(And(1/6 < y,\n",
    "                              y < 1/3),\n",
    "                          And(2/3 < y,\n",
    "                              y < 5/6)),\n",
    "                       1, \n",
    "                       0)\n",
    "\n",
    "    value = as_vector([gbar*(1 - (12*t)**2), 0])\n",
    "    bcs = [DirichletBC(W.sub(0), interpolate(value, V), (1, 2)),\n",
    "           DirichletBC(W.sub(0), zero(2), (3, 4))]\n",
    "    \n",
    "    a = (nu*inner(grad(u), grad(v)) - p*div(v) + q*div(u))*dx\n",
    "    L = inner(Constant((0, 0)), v)*dx\n",
    "    \n",
    "    wh = Function(W)\n",
    "    problem = LinearVariationalProblem(a, L, wh, bcs=bcs)\n",
    "    solver = LinearVariationalSolver(problem, solver_parameters=parameters, appctx={\"nu\": nu})\n",
    "    return solver"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "parameters = {\n",
    "    \"ksp_type\": \"gmres\",\n",
    "    \"ksp_monitor\": None,\n",
    "    \"pc_type\": \"fieldsplit\",\n",
    "    \"pc_use_amat\": True,\n",
    "    \"pc_fieldsplit_type\": \"schur\",\n",
    "    \"pc_fieldsplit_schur_fact_type\": \"lower\",\n",
    "    \"fieldsplit_0_ksp_type\": \"preonly\",\n",
    "    \"fieldsplit_0_pc_type\": \"mg\",\n",
    "    \"fieldsplit_1_ksp_type\": \"preonly\",\n",
    "    \"fieldsplit_1_pc_type\": \"python\",\n",
    "    \"fieldsplit_1_pc_python_type\": \"__main__.MassMatrix\",\n",
    "    \"fieldsplit_1_aux_pc_type\": \"icc\",\n",
    "}\n",
    "\n",
    "solver = create_solver(parameters)\n",
    "solver.solve()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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